Deciding if a given polynomial equation is solvable in radicals is a complex algorithmic task.
Can we instead list explicitly all types of polynomial equations of one unknown that are solvable by radicals?
Or all types of polynomial equations of one unknown that are not solvable by radicals?
[Ritt 1922] collects all rational functions of prime degree with inverses expressible in terms of radicals. So the corresponding polynomial equations are contained.
We get i. a. the following polynomial equations from Ritt's paper.
$a,b,c,d\in\mathbb{C}$
$m\in\mathbb{N}_{>0}$
1.)
$$a(z+b)^m+c=0$$
2.)
$T_m$: the m-th Chebyshev polynomial of the first kind
$$aT_m(bz+c)+d=0$$
$$a\left(2^{m-1}(bz+c)^m+\sum_{i=1}^{m-1}(-1)^i\frac{m}{i}\prod_{l=0}^{i-2}(m-i-1-l)\cdot 2^{m-2i-1}(bz+c)^{m-2i}\right)+d=0$$
Ritt continues with rational functions that are related to Weierstrass elliptic function $\wp$. Can they be listed explicitly?
Are there other schemes/systems/methods to list the requested types of polynomial equations explicitly?
$\ $